Adaptive GNSS positioning method capable of adjusting to ionospheric delay

ABSTRACT

A plurality of GNSS satellite signals feeds the signal processing engine operating in certain processing mode including carrier phase smoothed pseudorange positioning, precise point positioning (PPP), pseudorange differential (DGNSS), and carrier phase differential (RTK). The processing engine calculates two estimates of the ionosphere delay for each satellite: the filtered delay and the instant delay. Comparison of them allows to detect turbulent variation of the ionosphere and adjust parameters of two-parametric dynamic mode which improves positioning precision.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates to a satellite navigation systems, and,more particularly, to more accurate determination of ionospheric delaysin satellite navigation signals using an adjustable dynamic model.

Description of the Related Art

Assume that the receiver is capable of receiving multiple GNSS signalsof multiple GNSS satellites, including (but not limited by this set)

-   -   GPS L1, L2, L5;    -   Galileo L1, L2, E5a, E5b;    -   QZSS L1, L2, L5, E6;    -   GLONASS L1, L2, L3;    -   BEIDOU B1, B2, B3.

The number of such signals can exceed several dozen. The followingfundamental set of observables is used:

$\begin{matrix}{{P_{k,b}^{p}(t)} = {{\rho_{k}^{p}(t)} + {{cdt}_{k}(t)} - {{cdt}^{p}(t)} + {\left( \frac{f_{L\; 1}^{p}}{f_{b}^{p}} \right)^{2}{I_{k,{L\; 1}}^{p}(t)}} + {T_{k}^{p}(t)} + d_{k,b,P} + M_{k,b,P}^{p} - D_{b,P}^{p} + {ɛ_{k,b,P}^{p}(t)}}} & (1) \\{{\varphi_{k,b}^{p}(t)} = {{\frac{f_{b}^{p}}{c}{\rho_{k}^{p}(t)}} + {f_{b}^{p}{{dt}_{k}(t)}} - {f_{b}^{p}{{dt}^{p}(t)}} + {N_{k,b}^{p}\left( t_{{CS},k,b}^{p} \right)} - {\frac{1}{c}\frac{\left( f_{L\; 1}^{p} \right)^{2}}{f_{b}^{p}}{I_{k,{L\; 1}}^{p}(t)}} + {\frac{f_{b}^{p}}{c}{T_{k}^{p}(t)}} + d_{k,b,\varphi}^{p} + M_{k,b,\varphi}^{p} - D_{b,\varphi}^{p} + ɛ_{k,b,\varphi}^{p}}} & (2)\end{matrix}$

where the following notations are used (see [1, Chapter 6, 7]):

-   -   k is the index of the station;    -   p is the number of the satellite (out of K total satellites from        which signals are received). It is assumed that all satellites        available for tracking in the receiver station k are ordered and        this ordering number includes the satellite system. For example,        satellites with numbers from 1 to 5 are GPS, satellites with        numbers from 6 to 11 are GLONASS, satellites with numbers from        12 to 15 are BEIDOU, etc.;    -   b is the frequency band (for example L1, LP2, L2C, L5, E6, B1,        and so on);    -   (p,b) the pair of indices indicating the signal of the satellite        p at the frequency band b;    -   t is the current time; the term ‘epoch’ is also used to denote        the current discrete time instant;    -   c is the light speed;    -   f_(b) ^(p) is the frequency of the signal corresponding to the        satellite p and the frequency band b    -   dt_(k) (t) is the current clock bias of the station k;    -   dt^(p)(t) is the current clock bias of the satellite p;    -   I_(k,b) ^(p)(t) is the ionospheric delay affecting the signal        (p,b) received by the station k. Thus I_(k,L1) ^(p)(t) is        related to the L1 band. Basically, the ionospheric delay depends        on the position of the station, position of the satellite,        frequency of the signal, and the total electronic content (TEC)        corresponding to the time instant t;    -   T_(k) ^(p)(t) is the tropospheric delay. In contrast to the        ionospheric delay it doesn't depend on the signal frequency and        is called ‘non-dispersive’ delay.    -   P_(k,b) ^(p)(t) and φ_(k,b) ^(p)(t) are pseudorange and phase        measurements respectively;    -   M_(k,b,P) ^(p) and M_(k,b,φ) ^(p) are code and phase multipath,        affecting pseudorange and carrier phase measurements        respectively;    -   N_(k,b) ^(p)(t_(CS,k,b) ^(p)) carrier phase ambiguity        corresponding to the signal (p,b) received by the station k.        Note that it corresponds to the last detected cycle slip and        therefore it explicitly dependents on the time instant        t_(CS,k,b) ^(p), when the cycle slip was detected. The carrier        phase ambiguity remains unchanged until the cycle slip occurs.    -   ρ_(k) ^(p)(t) is the true topocentric range between the        satellite and the receiver station. The measurement        equations (1) and (2) relate this quantity with pseudorange and        carrier phase observables, respectively;    -   Quantities D_(b,φ) ^(p), D_(n,P) ^(p), ε_(k,b,P) ^(p)(t) and        ε_(k,b,φ) ^(p) denote hardware biases and noise. D stands for        satellite-related bias, while d denotes the receiver-related        bias. Biases reflect a systematic hardware component of the        measurement error. They are constant or slow varying, in        contrast to the noise component of the error having the        stochastic nature. Satellite position error is part of the slow        varying bias. It is also called ‘the ephemerides’ error.

Thus, the receiver position is measured by the pseudorange and carrierphase observables for the plurality of satellites. Error components,including biases and noise, affecting the observable equations (1) and(2), prevent a direct solution for the receiver antenna position.

Carrier phase measurements are much more precise, compared to thepseudorange measurements, since the carrier phase noise has standarddeviation in the centimeter or even millimeter range, while the standarddeviation of the pseudorange measurements is usually of the meter ordecimeter range. On the other hand, the carrier phase measurement isaffected by the carrier phase ambiguity, which is an unknown integervalued quantity.

Thus, elimination of measurement errors is necessary for precisepositioning. To achieve a high precision in position determination,different methods of errors mitigation are applied. For example,tropospheric errors can be precisely modeled and compensated inobservables of equations (1) and (2). Ionospheric errors can beestimated along with other unknowns. Noise is easily filtered.

Errors common to two receivers, like clock and hardware biases of thesatellite, can be compensated in a difference between two receivers.Usually one of receivers occupies a known position, while an antenna ofanother receiver is attached to the object to be located. The firstreceiver is called ‘the base’ while another receiver is called ‘therover’. The processing mode involving calculation of the across-receiverdifference (also called the ‘first difference’) is referred to asdifferential GNSS processing or DGNSS. The DGNSS processing is performedin real time and includes not only pseudoranges but also carrier phaseobservables, and is referred to as real time kinematic (RTK) processing.

Another sources of errors partially eliminated by across-receiverdifferences are ionospheric delay and ephemerides error. The closer therover is to the base, the better is compensation of the ionospheric andephemerides error.

For two stations k and l the across-receiver differences of pseudorangeand carrier phase measurements can be written as

$\begin{matrix}{{{{\overset{\_}{P}}_{{kl},b}^{p}(t)} = {{\rho_{k}^{p}(t)} - {\rho_{l}^{p}(t)} + {{cdt}_{kl}(t)} + {\left( \frac{f_{L\; 1}^{p}}{f_{b}^{p}} \right)^{2}{I_{{kl},{L\; 1}}^{p}(t)}} + d_{{kl},b,P} + {{\overset{\_}{ɛ}}_{{kl},b,P}^{p}(t)}}},} & (3) \\{{{{\overset{\_}{\varphi}}_{{kl},b}^{p}(t)} = {{\frac{1}{\lambda_{b}^{p}}\left( {{\rho_{k}^{p}(t)} - {\rho_{l}^{p}(t)}} \right)} + {f_{b}^{p}{{dt}_{kl}(t)}} + {N_{{kl},b}^{p}\left( t_{{CS},{kl},b}^{p} \right)} - {\frac{1}{\lambda_{b}^{p}}\left( \frac{f_{L\; 1}^{p}}{f_{b}^{p}} \right)^{2}{I_{{kl},{L\; 1}}^{p}(t)}} + d_{{kl},b,\varphi}^{p} + {\overset{\_}{ɛ}}_{{kl},b,\varphi}^{p}}},} & (4)\end{matrix}$

Another way for error mitigation includes using a precise satellitesclock and precise ephemerides. They are available through a dedicatedreal time service. Precise point positioning (PPP) allows to achieve thecentimeter level position with only one rover receiver, providedsatellite clock and precise satellite position are available. The basestation is not necessary in this case.

Finally, if neither base station, nor precise clock and ephemerides areavailable, the quality of the standalone position can be improved ifcarrier phase ambiguity and ionospheric delay are estimated, along withposition, using broadcast ephemerides. The corresponding processing modeis equivalent smoothing of pseudoranges using carrier phase measurementsor, in short, carrier phase smoothing of code pseudoranges, see [2].Whatever processing mode is used, the linearization and filteringalgorithms are used for recursive estimation of unknown position,carrier phase ambiguity, and ionospheric delay.

The general form of the linearized navigation model has the followingform (see [1, Chapt. 7])b _(P)(t)=Adx(t)+eξ(t)+Γi(t)+d _(P)  (5)b _(φ)(t)=Λ⁻¹ Adx(t)+Λ⁻¹ eξ(t)+n−Λ ⁻¹ Γi(t)+d _(φ)  (6)

Two last quantities are undifferences for carrier phase smoothing andPPP processing modes. For DGNSS and RTK processing modes the carrierphase ambiguity and ionospheric delay are across-receiver differenced.

Let n be total number of satellite signals, including differentsatellite systems, different satellites, different frequency bands.

In the following description, all vectors are represented by columns,and the superscript symbol ^(T) denotes the matrix transpose. R^(N) isthe N-dimensional Euclidean space. Given a linearization point x₀(t)∈R³,notations used in equations (5) and (6) are as follows:

-   -   b_(P)(t)∈R^(n) is the n-dimensional vector of pseudorange        residuals calculated at the linearization point;    -   b_(φ)(t)∈R^(n) is the n-dimensional vector of carrier phase        residuals calculated at the linearization point;    -   e=(1, 1, . . . , 1)^(T)∈R^(n) is the vector consisting of all        “ones”;    -   dx(t)∈R³ is the correction to the linearization point. Thus, the        corrected position is calculated as x(t)=x₀(t)+dx(t);    -   ξ(t) is the arbitrary varying rover clock shift; it is        undifferenced for the standalone and PPP processing modes, and        it is across-receiver differenced in the DGNSS and RTK        processing modes;    -   Λ is the n-dimensional diagonal matrix with wavelengths λ_(b)        ^(p)=c/f_(b) ^(p) in the main diagonal. Each wavelength        corresponds to the specific signal (p,b);    -   A is the n×3 matrix of directional cosines;    -   Γ is the n×n diagonal matrix with quantities (f_(L1) ^(p)/f_(b)        ^(p))² in the main diagonal;    -   i(t)∈R^(n) is the vector of ionospheric delays related to the L1        frequency band. Ionospheric delays are undifferenced for the        standalone and PPP processing modes, and they are        across-receiver differenced in the DGNSS and RTK processing        modes;    -   n∈R^(n) is the vector of carrier phase ambiguities related to        the L1 frequency band. Ambiguities are undifferenced for the        standalone and PPP processing modes, and they are        across-receiver differenced in the DGNSS and RTK processing        modes;    -   d_(P)∈R^(n) and d_(φ)∈R^(n) are vectors of pseudorange and        carrier phase receiver hardware biases.

Consideration of pseudorange hardware biases leads to a necessity toconsider the plurality of signals the receiver is able to track. In thecase of a multi-frequency and multi-system receiver supporting thefollowing bands:

L1, L2 and L5 bands for GPS,

L1 and L2 GLONASS,

L1, E5a, E5b and E6 Galileo,

L1, L2, L5 and E6 QZSS,

L1 an, L5 SBAS,

B1, B2, and B3 Beidou,

the signals (L1 GPS, L1 Galileo, L1 SBAS, L1 QZSS), (L2 GPS, L2 QZSS),(L5 GPS, E5a Galileo, L5 SBAS, L5 QZSS), (E6 Galileo, E6 QZSS),respectively, can share the same hardware channel and therefore will beaffected by the same hardware bias, as noted in [1, Chapter 7]. Notethat the biases vector d_(P) and the clock shift variable ξ(t) appear asa sum in equation (5). This means that one of the biases, say d_(L) ₁_(,G/E/S/Q,P), can be combined with ξ(t), while others can be replacedwith their difference with d_(L) ₁ _(,G/E/S/Q,P) . Thus, new biasvariables appear:η₁ =d _(L) ₂ _(,G/Q,P) −d _(L) ₁ _(,G/E/S/Q,P),η₂ =d _(L) ₁ _(,R,P) −d_(L) ₁ _(,G/E/S/Q,P),η₃ =d _(L) ₂ _(,R,P) −d _(L) ₁ _(,G/E/S/Q,P)η₄ =d _(L) ₅ _(,G/E/S/Q,P) −d _(L) ₁ _(,G/E/S/Q,P),η₅ =d _(E) _(5b)_(,E,P) −d _(L) ₁ _(,G/E/S/Q,P),η₆ =d _(E) ₆ _(,E/Q,P) −d _(L) ₁_(,G/E/S/Q,P)η₇ =d _(B) ₁ _(,B,P) −d _(L) ₁ _(,G/E/S/Q,P),η₈ =d _(B) ₂ _(,B,P) −d_(L) ₁ _(,G/E/S/Q,P),η₉ =d _(B) ₃ _(,B,P) −d _(L) ₁ _(,G/E/S/Q,P)  (7)

This representation can be referred to as establishing a bias datum.

In one possible embodiment, the linearized equations (5) can now beexpressed in the formb _(p)(t)=Adx(t)+eξ(t)+Γi(t)+W _(η)η  (8)

The bias vector η has the appropriate dimension m_(η). Note again thatin this exemplary embodiment we follow notations introduced in [1],which is incorporated herein by reference in its entirety.

The bias vector η is three-dimensional (m_(η)=3) for dual-band anddual-system GPS/GLONASS receivers, as only biases η₁, η₂, η₃ arepresented among all possible biases listed in (7). In the case of themulti-band, multi-system receiver, the dimension of the vector η can belarge. It is one-dimensional in the case of dual-band GPS-only receiversor single band GPS/GLONASS receivers.

The W_(η) is referred to as bias allocation matrix and has dimensionsn×m_(η). It allocates a single bias, or none, to a certain signal. Nobias is allocated to the signal corresponding to the GPS, Galileo, SBAS,or QZSS systems and b=L1 because we combined the bias d_(L1,G/E/S/Q,P)with the clock bias ξ(t). In this case, the row of W_(η) consists ofzeroes.

Consider, for example, a dual-band GPS/GLONASS receiver. Suppose ittracks six GPS satellites and six GLONASS satellites. The total numberof dual-band signals is 24. Let the signals be ordered in the followingway: six GPS L1, six GPS L2, six GLONASS L1, and six GLONASS L2 signals.The biases allocation matrix presented in the linearized singledifference pseudorange equation (8) takes the following form:

$\begin{matrix}{W_{\eta}^{T} = \left\lbrack \begin{matrix}0 & 0 & \; & 0 \\0 & 0 & \vdots & 0 \\0 & 0 & \; & 0\end{matrix} \middle| \begin{matrix}1 & 1 & \; & 1 \\0 & 0 & \vdots & 0 \\0 & 0 & \; & 0\end{matrix} \middle| \begin{matrix}0 & 0 & \; & 0 \\1 & 1 & \vdots & 1 \\0 & 0 & \; & 0\end{matrix} \middle| \begin{matrix}0 & 0 & \; & 0 \\0 & 0 & \vdots & 0 \\1 & 1 & \; & 1\end{matrix} \right\rbrack} & (9)\end{matrix}$

Further, the real-valued carrier phase ambiguities (also called floatambiguities) are combined with biases d_(φ)), while pseudorange biasesare considered as a real valued constant unknown parameter. Thus, aftercombination of carrier phase ambiguities with carrier phase biases, theequation (6) takes the formb _(φ)(t)=Γ⁻¹ Adx(t)+Γ⁻¹ eξ(t)+n−Λ ⁻¹ Γi(t)  (10)

Note that the noise component is omitted in equations (8), (10) for thesake of brevity.

Accordingly, there is a need in the art for a more accuratedetermination of phase to, thereby, enable more accurate determinationof position.

SUMMARY OF THE INVENTION

Accordingly, the present invention is directed to automatic adjusting ofparameters of the ionosphere dynamic model to the current state of theionosphere in order to achieve better precision of positioning in thefollowing processing modes:

Standalone with carrier phase smoothing of pseudoranges,

PPP,

DGNSS,

RTK.

The same ionosphere delay and carrier phase ambiguity described above isused in all processing modes. Adaptive adjustment of parameters isperformed after the recursive update step according to the equation (19)(described in the following text) is performed.

Additional features and advantages of the invention will be set forth inthe description that follows, and in part will be apparent from thedescription, or may be learned by practice of the invention. Theadvantages of the invention will be realized and attained by thestructure particularly pointed out in the written description and claimshereof as well as the appended drawings.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and areintended to provide further explanation of the invention as claimed.

BRIEF DESCRIPTION OF THE ATTACHED FIGURES

The accompanying drawings, which are included to provide a furtherunderstanding of the invention and are incorporated in and constitute apart of this specification, illustrate embodiments of the invention andtogether with the description serve to explain the principles of theinvention.

In the drawings:

FIG. 1 illustrates constellation of GNSS satellites, the GNSS antennaand receiver, and signals travelling from satellites to the antennathrough the ionospheric layer.

FIG. 2 illustrates algorithm of the adaptation of the time constantτ_(i). and the scale factor s.

FIG. 3 illustrates an exemplary navigation receiver on which theinvention may be implemented.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

Reference will now be made in detail to the embodiments of the presentinvention, examples of which are illustrated in the accompanyingdrawings.

The vector of ionospheric delays i(t) is supposed to be slow varying.Its time variation is subjected to a certain dynamic model. To specifythe dynamic model that governs variation of i(t) let us suppose thattime variation of the vector i(t) is restricted by equationsi ^(p)(t)=γ_(i) i ^(p)(t−1)+ε_(i)(t)  (11)

where

$\begin{matrix}{\gamma_{i} = e^{- \frac{\Delta\; t}{\tau_{i}}}} & (12)\end{matrix}$

with Δt being the across-epoch time difference, and τ_(i) being the timeconstant reflecting the rate of variation of the ionospheric delay intime. A typical value for τ_(i) is 600-1800 seconds, provided noionospheric disturbances and scintillation occurs. Otherwise, ifdetection criteria described below indicate a presence of fast variationof ionosphere (more precisely, a variation of content of chargedparticles, or total electron content (TEC) in the ionosphere), theinstant value of τ_(i) can be temporary scaled down until the fastvariations end. Moreover, the algorithm presented here is able toadaptively adjust the constant τ_(i) to the rate of real variation ofthe ionosphere state.

The white noise ε_(t) (t) has a variance σ_(i) ² which provides thevariance of the resulting ionosphere delay generated by the differenceequation (11). For the case of RTK and DGNSS processing modes theresulting residual across-receiver delay is supposed to satisfy thecondition∥i _(p)(t)|≈s×10⁻⁶×∥baseline length∥  (13)

with a certain scale factor s, which takes values of, for example, from1 to 5, depending on the current solar activity (for a period withunusually high solar activity, a value of s up to 10 might be used). Forthe standalone and PPP processing modes it should satisfy the condition∥i ^(p)(t)|≈s×σ _(ref)  (14)

where σ_(ref) is a certain reference value corresponding to the averagedstate of the ionosphere, s is a scale factor having the same meaning asin (13).

It follows from (11) that ∥i^(p)(t)∥²=∥γ_(i)i^(p)(t−1)+ε_(i)(t)∥². Then,assuming that the stochastic process i^(p)(t) is stationary and i^(p)(t)does not depend on ε_(i)(t), we take a mean value of both sides of thelast equality. We obtainσ_(i) ²=(1−γ_(i) ²)×(s×10⁻⁶×∥baseline length∥)²  (15)orσ_(i) ²=(1−γ_(i) ²)×(s×σ _(ref))²  (16)

depending on the processing mode. Having in mind application of thefiltering scheme for estimation of variables listed in equations (8) and(10), let us define the measurement and dynamic models. The measurementmodel combines equations (8) and (10). The group of variables η and ηform the vector

$\begin{matrix}{y = \begin{pmatrix}\eta \\n\end{pmatrix}} & (17)\end{matrix}$

The group of arbitrary varying variables dx(t) and ξ(t) form the vector

$\begin{matrix}{{\overset{\_}{x}(t)} = \begin{pmatrix}{{dx}(t)} \\{\xi(t)}\end{pmatrix}} & (18)\end{matrix}$

The group of slow varying variables is arranged into the vector i(t)governed by the dynamic model (11). Then the estimation algorithmdescribed in [1, Table 3.5.1 and Chapt. 7.6, 7.7] can be applied.

Taking into account notations (17) and (18) the recursive estimationalgorithm takes the following form

$\begin{matrix}{\begin{pmatrix}{i\left( {t + 1} \right)} \\{y\left( {t + 1} \right)}\end{pmatrix} = {\begin{pmatrix}{\gamma_{i}{i(t)}} \\{y(t)}\end{pmatrix} + {G_{iy}{r^{\prime}\left( {t + 1} \right)}}}} & (19) \\{{\overset{\_}{x}\left( {t + 1} \right)} = {{G_{x}\left( {t + 1} \right)}{r^{''}\left( {t + 1} \right)}}} & (20)\end{matrix}$

with G_(iy) and G_(x) being the matrix gain, r′(t+1) and r″(t+1) beingthe residuals of the linearized measurement equations (8) and (10)calculated before and after the update (19). The matrix gain iscalculated according to the algorithm described in [1, Chapters 7.6,7.7]. We present the full description here for the sake of completeness:

$\begin{matrix}{G_{iy} = {\left( M^{T} \right)^{- 1}{M^{- 1}\begin{pmatrix}H^{T} \\W^{T}\end{pmatrix}}\Pi}} & (21) \\{G_{x} = {\left( L_{J}^{T} \right)^{- 1}\left( L_{J} \right)^{- 1}J^{T}}} & (22)\end{matrix}$

where:

-   -   J^(T)J=L_(J)L_(J) ^(T), calculation of the Cholesky        factorization;    -   Π=I_(2n)−J(L_(J) ^(T))⁻¹(L_(J))⁻¹J^(T);

I_(n) is the identity matrix, 0 is the zero matrix of the appropriatesize;

-   -   F=γ_(i)I_(n);    -   Q=σ_(i) ² I_(n);

To complete description, let us introduce the recursive scheme with thematrix {circumflex over (D)}(t)∈R^(K+n) ^(y) being recursively updated,{circumflex over (D)}(0)=0, n_(y) is dimension of the vector y,

Last equation (26) defines the Cholesky factorization. Finally.{circumflex over (D)}(t+1)=MM ^(T)  (27)

The matrix M is used in equation (21).

Let us define also the filtering reset procedure. If for any reason the‘filtering reset’ decision is taken at the time instant t+1, the matrix{circumflex over (D)}(t) is set to zero:{circumflex over (D)}(t):=0  (28)

which means that the filtering scheme is started from the scratch at thetime instant t+1.

Let us define also the partial filtering reset procedure with the fadingfactor α<1. It means that the memory accumulated in the matrix{circumflex over (D)}(t) is partially ‘forgotten’:{circumflex over (D)}(t):=α{circumflex over (D)}(t)  (29)

It should be noted that for the sake of brevity we omitted handling ofcovariance matrices of the pseudorange and carrier phase noise.Moreover, we omitted this noise component in the basic equations (5) and(6).

The main problem with application of this straightforward numericalscheme is that the two parameters τ_(i) and s presented in the inequations (11), (12), and (13), reflecting the current state of theionosphere, are usually unknown. Moreover, the dynamic model (11) isapplicable to the stationary, time invariant, “calm” state of theionosphere. In practice, the ionosphere is subject to turbulence, whichmakes the model (11) with constant parameters τ_(i) and s inapplicable.The current invention deals with the adaptive algorithm being able toadjust variation of the model parameters to the time varying state ofthe ionosphere.

Assuming that constant parameters (17) are updated at the time instantt+1 by the recursive scheme (19), the current estimate of the ambiguityvector n(t+1) is available. Assuming that the number of satellites K isnot less than 4, the number of frequency bands is not less than 2, andthe number of signals n is not less than 8, the equation (10) can besolved with respect to the variables dx(t+1), ξ(t+1), i(t+1). Let usdenote these instant estimates by dx*(t+1), ξ*(t+1), i*(t+1). Then:

$\begin{matrix}{\begin{pmatrix}{{dx}^{*}\left( {t + 1} \right)} \\{\xi^{*}\left( {t + 1} \right)} \\{i^{*}\left( {t + 1} \right)}\end{pmatrix} = {\left( {{\overset{\_}{A}}^{T}\overset{\_}{A}} \right)^{- 1}{{\overset{\_}{A}}^{T}\left( {{b_{\varphi}\left( {t + 1} \right)} - {n\left( {t + 1} \right)}} \right)}}} & (30)\end{matrix}$

whereĀ=[Λ ⁻¹ A|Λ ⁻¹ e|−Λ ⁻Γ]  (31)

Thus two estimates of the ionosphere delay vector i are available at thetime instant t+1:

-   -   The filtered estimate i(t+1) calculated through the update        equation (19) taking into account the dynamic model (11) with        parameters τ_(i) and s;    -   The instant estimate i*(t+1) calculated based on instant carrier        phase measurements and instant carrier phase ambiguity estimate        n(t+1), not taking into account the dynamic model (11).

Comparison of these two estimates allows to judge whether the ionospheremodel with currently used parameters τ_(i) and s is applicable. One ofthe criteria proposed in this discussion indicates that the parameterτ_(i) should be downscaled. Another criterion indicates that theparameter s should be increased or decreased.

A large body of literature is devoted to detection of fast variation ofthe ionospheric delays, mainly in connection with scintillationdetection. The standard approach consists of analysis of time series foreach satellites individually. Two criteria S₄ and σ_(φ) (amplitude andphase scintillation indices, respectively) are used. Various decisionrules, like Neyman-Pearson (see [2] and references cited therein) can beapplied to generate binary criteria.

A well known approach to mitigation of the ionosphere delay consists inconstruction of the ionosphere-free combination (IFC) of pseudorange andcarrier phase measurements (see for example [1]). An advantage of usingIFC is that it allows for almost total rejection of the ionospheredelay. A disadvantage is that it does not take into account slow varyingnature of the ionosphere delay. Moreover, using IFC increasessensitivity of the resulting position to the measurement's noise anderrors of another nature. The method of adjusting parameters of IFC isconsidered in [3].

The approach described in the present discussion does not usetime-series analysis for each satellite individually. Instead, it usesthe criteria in which all satellites are involved simultaneously. It canbe stated that the current invention uses the state space domainanalysis instead of time domain analysis, allowing for instant detectionof ionospheric turbulence. For the case of calm ionosphere, it allows toproduce less noisy position, compared to IFC-based approaches. For thetime varying turbulent ionosphere, it allows to adjust parameters of thedynamic model (11).

FIG. 2 shows detailed flow chart of the proposed algorithm of adaptiveadjustment of parameters τ_(i) and s. After filtering (see block 201),the instant ionospheric delay is calculated (see block 202). Having athand two estimates i(t+1) and i*(t+1), the correlation coefficientbetween them is calculated as (see block 203):

$\begin{matrix}{{c\left( {t + 1} \right)} = \frac{\left\langle {{i\left( {t + 1} \right)},{i^{*}\left( {t + 1} \right)}} \right\rangle}{{{i\left( {t + 1} \right)}}{{i^{*}\left( {t + 1} \right)}}}} & (32)\end{matrix}$

where

a, b

denotes the scalar product of the two vectors a and b; ∥a∥ denotes theEuclidean norm of the vector a. Both vectors i(t+1) and i*(t+1) have thedimension K, i.e., the number of satellites.

A high correlation between two estimates (see block 208) means that theparameter τ_(i) is most likely chosen correctly and there is no need tochange it (see block 209). Otherwise, a low value of the correlationcoefficient c(t+1) (less than the threshold value T_(T)) shows thatτ_(i) should be decreased (see block 204) allowing the value i(t+1) tochange faster. Let S_(down)<1 be the downscaling factor. In oneembodiment T_(l) can be chosen 0.9 (for example) and S_(down) can bechosen 0.9 (for example). Then the parameter τ_(i) is updatedτ_(i):=τ_(i) ×S _(down)  (33)

Moreover, the filtering algorithm (19), (21)-(27) (see block 201) can betotally or partially reset according to equations (28) or (29) with α<1being the parameter of the algorithm. In one embodiment α can be chosenα=0.1. The total reset is a particular case of (29) with α=0.

In the case of the partial reset, the current estimate i(t+1) is updated(see block 204) according to the equationi(t+1):=αi(t+1)+(1−α)i*(t+1)  (34)

If c(t+1) is above the upper threshold value T_(u) (0<T_(l)<T_(u)<1)then τ_(i) should be increased. Let S_(up)>1 be the magnificationfactor. In one embodiment T_(u) can be chosen 0.99 and S_(up)=1.1. Theparameter τ_(i) is updated (see block 205) according toτ_(i):=min {τ_(i) ×S _(up),τ*}  (35)

with updated value not exceeding the pre-defined value τ* which can bechosen as 1200 seconds (for example).

If T_(l)<c(t+1)<T_(u) then τ_(i) is kept unchanged.

Even if the correlation is good, the values ∥i*(t+1)∥ and ∥i(t+1)∥ arenot necessary nearly equal. If, for example (see block 210), theinequality∥i*(t+1)∥>∥i(t+1)∥  (36)

holds for several sequential time instances (say, 5), then the value ofs should be increased (see block 206):S:=s×S _(up)  (37)

Otherwise (see block 211), if the inequality∥i*(t+1)∥<∥i(t+1)∥  (38)

holds for several sequential time instances (say, 5), then the value ofs should be decreased (see block 207):s:=s×S _(down)  (39)

The values of S _(up) and S _(down) can be chosen 1.1 and 0.9,respectively.

FIG. 3 illustrates an exemplary navigation receiver on which theinvention may be implemented. As will be appreciated by one of ordinaryskill in the art, the various blocks and components shown in FIG. 3 canbe implemented as discrete components, as an ASIC or multiple ASICs oras software running on a processor.

Having thus described the different embodiments of a system and method,it should be apparent to those skilled in the art that certainadvantages of the described method and apparatus have been achieved. Itshould also be appreciated that various modifications, adaptations, andalternative embodiments thereof may be made within the scope and spiritof the present invention. The invention is further defined by thefollowing claims.

REFERENCES (INCORPORATED HEREIN BY REFERENCE IN THEIR ENTIRETY)

-   1. A. Leick, L. Rapoport, D. Tatarnikov, GPS Satellite Surveying,    Wiley & Sons (2015)-   2. D. V. Ratnam, G. Sivavaraprasad, J. Lee, Automatic ionospheric    scintillation detector for global navigation satellite system    receivers, IET Radar, Sonar & navigation, DOI:    10.1049/iet-rsn.2014.0232 (February 2015).-   3. S.V. Averin, R.A. Nevzorov, A.V. Plenkin, V.I. Zubinsky,    Mitigation of scintillations in signals of global navigation    satellite systems caused by ionospheric irregularities, U.S. Pat.    No. 9,244,174, issued Jan. 26, 2016.

What is claimed is:
 1. A satellite navigation positioning method,comprising: receiving n navigation signals from K navigation satellites,wherein the n navigation signals are on at least two different carrierfrequencies, and wherein n is at least 8 and K is at least 4; on aprocessor of a navigation receiver, calculating current coordinatesbased on the n navigation signals; executing an adaptive positioningalgorithm for adjusting parameter z representing a time constant of arate of variation of ionospheric delay, and a scale factor s, whichranges from 1 to 10 and which relates to a baseline length and solaractivity; and correcting the calculated current coordinates based on theadaptive positioning algorithm.
 2. The method of claim 1, furthercomprising calculating a filtered estimate i(t+1) using parameters τ_(i)and s.
 3. The method of claim 2, further comprising calculating aninstant estimate i*(t+1) calculated based on instant carrier phasemeasurements and instant carrier phase ambiguity estimate.
 4. The methodof claim 3, further comprising reducing τ_(i) if the correlationcoefficient is less than a threshold value T_(l).
 5. The method of claim3, further comprising increasing τ_(i) if the correlation coefficient isgreater than a threshold value T_(l).
 6. The method of claim 2, furthercomprising calculating a correlation coefficient${c\left( {t + 1} \right)} = {\frac{\left\langle {{i\left( {t + 1} \right)},{i^{*}\left( {t + 1} \right)}} \right\rangle}{{{i\left( {t + 1} \right)}}{{i^{*}\left( {t + 1} \right)}}}.}$7. The method of claim 1, further comprising increasing s if∥i*(t+1)∥>∥(t+1)∥ is true for five sequential time instances.
 8. Themethod of claim 1, further comprising decreasing s if∥i*(t+1)∥<∥i(t+1)∥ is true for five sequential time instances.